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Given that
u=3w^(4)-5, find
(d)/(dw)(4w^(2)-sin u) in terms of only
w.
Answer:

Given that $u=3 w^{4}-5$ , find $\frac{d}{d w}\left(4 w^{2}-\sin u\right)$ in terms of only $w$ . $\newline$ Answer:

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Write a formula for an arithmetic sequence

Full solution

Q. Given that

u=3 w^{4}-5

, find

\frac{d}{d w}\left(4 w^{2}-\sin u\right)

in terms of only

w

.

\newline

Answer:

Identify Function and Rule: We need to find the derivative of the function $4w^2 - \sin(u)$ with respect to $w$ . To do this, we will use the chain rule, which states that the derivative of a composite function is the derivative of the outer function times the derivative of the inner function. In this case, the outer function is $4w^2 - \sin(x)$ and the inner function is $u = 3w^4 - 5$ .
Differentiate Outer Function: First, we differentiate the outer function with respect to $u$ , treating $w$ as a constant. The derivative of $4w^2$ with respect to $u$ is $0$ , since $w$ is treated as a constant and the derivative of $-\sin(u)$ with respect to $u$ is $-\cos(u)$ .
Differentiate Inner Function: Next, we differentiate the inner function $u = 3w^4 - 5$ with respect to $w$ . The derivative of $3w^4$ with respect to $w$ is $12w^3$ , and the derivative of $-5$ with respect to $w$ is $0$ . So, the derivative of $u$ with respect to $w$ is $12w^3$ .
Apply Chain Rule: Now, we apply the chain rule by multiplying the derivative of the outer function with respect to $u$ by the derivative of the inner function with respect to $w$ . This gives us $(0 - \cos(u)) \times 12w^3$ .
Simplify Expression: Simplify the expression by distributing the $12w^3$ across the terms inside the parentheses. This results in $-12w^3 \cdot \cos(u)$ .
Express Derivative in Terms of $w$ : Finally, we need to express the derivative in terms of $w$ only. Since $u = 3w^4 - 5$ , we substitute $u$ back into the expression $-12w^3 \cdot \cos(u)$ to get $-12w^3 \cdot \cos(3w^4 - 5)$ .

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